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Consider the linear model

$$y_t= x_t'\beta +\epsilon_t$$ for $t=1,...,T$.

where $x_t= ( x_{1t} \ \ x_{2t} \ \ ... \ \ x_{kt}) $ and $\beta$ is $(k\times 1)$ vector of unknown coefficients.

Given $z_t= ( z_{1t} \ \ z_{2t} \ \ ... \ \ z_{ht})' $ with $E(z_t) \not= 0$ for $h>k$ is a white noise disturbance term

with $E(z_t\epsilon_t)=0 \ \ \forall t$.

Also, a symmetric and positive definite weight matrix $S_n$ is defined.

Firstly, we need to find the Generalized Method of Moments (GMM) estimator of $\beta$.

Secondly, we need to show that $\hat{\beta}_{GMM}$ is a consistent estimator of $\beta$ for $plim(\frac{1}{T} \sum z_t x_t')= R$ and $plim(S_n)=S$ for constant and finite matrices $R$ and $S$.

Thirdly, we need to show the reliability of this estimator $\hat{\beta}_{GMM}$ (with mathematical expressions and calculations) when the linear model is misspecified with $\epsilon_t = \gamma T^{-1/2} + e_t$ where $e_t$ is iid disturbance with $E(e_t)=0$


What I did is that...

I did the first and second parts.

I find the Generalized Method of Moments (GMM) estimator of $\beta$ as

$$\hat{\beta}_{GMM} = (X'ZS_nZ'X)^{-1}X'ZS_nZ'y$$

As for the second part,

$$\hat{\beta}_{GMM} = (X'ZS_nZ'X)^{-1}X'ZS_nZ'(X\beta +\epsilon)$$

$$\hat{\beta}_{GMM} = \beta +(X'ZS_nZ'X)^{-1}X'ZS_nZ'\epsilon$$

$$plim(\hat{\beta}_{GMM}) = \beta +(plim(\frac{X'Z}{T})plim(S_n)plim(\frac{Z'X}{T}))^{-1}plim(\frac{X'Z}{T})plim(S_n)plim(\frac{Z'\epsilon}{T})$$

$$plim(\hat{\beta}_{GMM}) = \beta + (R'SR)^{-1}R'S plim(\frac{Z'\epsilon}{T})$$

since $plim(\frac{Z'\epsilon}{T})=0$ by law of large numbers, $plim(\hat{\beta}_{GMM}) = \beta$


Mu actual question is the third part in the gray box. I could not show whether $\hat{\beta}_{GMM} $ is reliable or not under the given conditions.

I think that the GMM estimator is reliable if the estimator is consistent and efficient. But I could not proceed this. And I could not show this third part.

what I did for the part 3

Population moment condition requires $E(z_t\epsilon_t)=0$

If this requirement holds, then the estimator $\hat{\beta}_{GMM} $ will be reliable.

We know that $E(z_t)=\mu_z\not= 0$

$$E(z_t \epsilon_t)=E(z_t(\gamma T^{-1/2}z_t z_te_t))$$

$$= \gamma T^{-1/2}E(z_t)+E( z_te_t)= \gamma T^{-1/2}\mu_z + E( E(z_te_t|z_t))= \gamma T^{-1/2}\mu_z + E(z_t E(e_t|z_t))= \gamma T^{-1/2}\mu_z +0 $$

where $E(e_t|z_t)=0$ since $e_t$ is i.i.d and $E(e_t)=0$

So, $$E(z_t \epsilon_t)= \gamma T^{-1/2}\mu_z\not=0$$ for $\gamma\not=0$. In this case, the GMM estimation is not reliable. On the other hand, if $\gamma=0$, then the GMM estimator becomes reliable.

What do you think about this solution? Or what is your idea on how to show this question? I will be glad if you help me. Thank you.

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